This is a bit of a complicated issue. If one parent has full custody, that parent is responsible for the care and maintenance of that child, or children. As far as simply dropping off that child with the other parent, that other parent has to consent, or it has to be so stated in the custodial order. If neither of those exist, the other parent can refuse, or can go to court and file for emergency sole custody, and use that as a basis to terminate the child support obligation. Until the order is terminated the duty to pay support still exists,however--the courts have to terminate before one parent can stop paying.